Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

D. Andreucci; M.A. Herrero; J.J.L. Velázquez

doi:10.1016/s0294-1449(97)80148-5

JournalsaihpcVol. 14, No. 1pp. 1–53

Liouville theorems and blow up behaviour in semilinear reaction diffusion systems

D. Andreucci
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università La Sapienza, via A. Scarpa 16, 00181 Roma, Italy
M.A. Herrero
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid Spain
J.J.L. Velázquez
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, 28040 Madrid Spain

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Abstract

This paper is concerned with positive solutions of the semilinear system:

{u_{t} = Δ u + υ^{p}, υ_{t} = Δ υ + u^{q}, p \geq 1, q \geq 1, (S)

which blow up at $x = 0$ and $t = T < \infty$ . We shall obtain here conditions on $p$ , $q$ and the space dimension $N$ which yield the following bounds on the blow up rates:

u (x, t) \leq C (T - t)^{- \frac{p + 1}{pq - 1}}, υ (x, t) \leq C (T - t)^{- \frac{q + 1}{pq - 1}}, (1)

for some constant $C > 0$ . We then use (1) to derive a complete classification of blow up patterns. This last result is achieved by means of a parabolic Liouville theorem which we retain to be of some independent interest. Finally, we prove the existence of solutions of (S) exhibiting a type of asymptotics near blow up which is qualitatively different from those that hold for the scalar case.

Cite this article

D. Andreucci, M.A. Herrero, J.J.L. Velázquez, Liouville theorems and blow up behaviour in semilinear reaction diffusion systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 1, pp. 1–53

DOI 10.1016/S0294-1449(97)80148-5